| L(s) = 1 | − 2-s + 0.445·3-s + 4-s + 5-s − 0.445·6-s − 3.24·7-s − 8-s − 2.80·9-s − 10-s + 1.60·11-s + 0.445·12-s + 3.24·14-s + 0.445·15-s + 16-s + 3.10·17-s + 2.80·18-s − 2.89·19-s + 20-s − 1.44·21-s − 1.60·22-s + 6.78·23-s − 0.445·24-s + 25-s − 2.58·27-s − 3.24·28-s + 4.04·29-s − 0.445·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.256·3-s + 0.5·4-s + 0.447·5-s − 0.181·6-s − 1.22·7-s − 0.353·8-s − 0.933·9-s − 0.316·10-s + 0.483·11-s + 0.128·12-s + 0.867·14-s + 0.114·15-s + 0.250·16-s + 0.754·17-s + 0.660·18-s − 0.663·19-s + 0.223·20-s − 0.315·21-s − 0.341·22-s + 1.41·23-s − 0.0908·24-s + 0.200·25-s − 0.496·27-s − 0.613·28-s + 0.751·29-s − 0.0812·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.445T + 3T^{2} \) |
| 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 - 1.60T + 11T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 - 6.78T + 23T^{2} \) |
| 29 | \( 1 - 4.04T + 29T^{2} \) |
| 31 | \( 1 + 8.31T + 31T^{2} \) |
| 37 | \( 1 - 3.20T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 5.18T + 43T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 5.28T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 7.50T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 6.05T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.072901277198171654677132001576, −8.388003087986880720697198289981, −7.38564040461661719535119942861, −6.51535740340499805263126404481, −6.02138033230657013565327411152, −4.95423418070632430006890707802, −3.38297194691024270094409252993, −2.94742121664306434268952366890, −1.59241437633239548266475978420, 0,
1.59241437633239548266475978420, 2.94742121664306434268952366890, 3.38297194691024270094409252993, 4.95423418070632430006890707802, 6.02138033230657013565327411152, 6.51535740340499805263126404481, 7.38564040461661719535119942861, 8.388003087986880720697198289981, 9.072901277198171654677132001576
